WASHINGTON (USA Today) -The Senate's top two leaders have asserted control over budget negotiations with the White House for a deal to reopen the federal government and avoid a default on the nation's debt.

"I hope that our talking is some solace to the American people and to the world," Reid said. "This should be seen as something very positive, even though we don't have anything done yet."

Senate Democratic leaders will meet Saturday afternoon at the White House with President Obama.

The momentum has shifted to the Senate as it became increasingly clear that House Speaker John Boehner, R-Ohio, was unable to strike a deal with President Obama on his own.

Following a closed-door House GOP meeting Saturday morning, lawmakers acknowledged a diminished role to resolve the impasse.

"I'm disappointed that the president has rejected the offer that we put on the table. I know that he's trying to see which Republican senator he can pick off in the Senate. I hope that the Senate Republicans stand strong so we can speak with one voice," said House Majority Leader Eric Cantor, R-Va., after the meeting.

Reid rejected a bipartisan effort led by Sen. Susan Collins, R-Maine, to end the shutdown and lift the debt ceiling because he said Democrats should not have to make concessions in order for Congress to reopen government and avoid a first-ever national default. Collins' plan included reforms to the health care law and locked in federal spending levels Democrats oppose.

"I appreciate her efforts as always to find a consensus, but the plan that she's suggested that I've seen in writing is not going to go anyplace at this stage," Reid said. "They're not doing (Democrats) a favor by reopening the government. They're not doing us a favor by extending the debt ceiling. That's part of our jobs."

It is unclear what budget framework can pass a Democratic-controlled Senate and a GOP-controlled House, where Boehner is under significant political pressure from conservatives to deliver a package that reins in the health care law in some way, or sets up a path for significant fiscal reforms.

Senate Democrats made clear Saturday that they were confident that House Republicans would eventually advance what the Senate can pass, even if it means Boehner will need to rely on House Democrats to do so. Democrats only want to negotiate on long-term budget issues after the government is reopened and the threat of default is off the table.

"We are not over-playing our hand. We are open to discussion in every way, and I think our Republican colleagues are moving in our direction," said Sen. Charles Schumer, D-N.Y., "The fact that Obamacare is not a major part of the discussion anymore among most all Senate and many House Republicans all shows it's moving in a decent direction."

Schumer said Democrats were open, however, to negotiations to alleviate unpopular, across-the-board spending cuts known as the sequester, and that the cuts were one of the central issues in the ongoing talks.

The government shutdown, which entered its 12th day Saturday, began when House Republicans refused to advance a stopgap funding bill to keep government open unless it dismantled the president's health care law in some way.

The shutdown has rolled in to a battle to increase the nation's borrowing authority by Oct. 17, or risk a U.S. default on its debts that could rattle financial markets and cause sweeping economic unrest.

Obama told business leaders on a conference call Friday that he hoped a deal could be struck this weekend, but lawmakers were not optimistic that a final deal would be reached this weekend. Schumer said he expected a result in "the next few days" which means Congress could go right up against the default date.

Reid emphasized Saturday that it could be catastrophic for financial markets to reopen with no deal done and the threat of U.S. default looming. "Everyone will lose. Not only in America, but around the world."